In [1]} it is shown that there exists an explicitly computable constant \(\mathfrak {f}_{V,\varepsilon }\)  for such that for all hyperbolic rational homology sphere \(Y\)  with \(\mathrm {vol}\le V\) , \(\mathrm {inj}\ge \varepsilon \) , we have the inequality
\(|\mathrm {gr}^{\mathbb {Q}}([\mathfrak {b}_0])|\le \mathfrak {f}_{V,\varepsilon }\) 
for the absolute grading of the first boundary stable critical point \([\mathfrak {b}_0]\) . Notice that in [2]} the assumption that \(\lambda _1^*>2\)  implies that the operator \(D_{B_{\bullet }}\)  has no kernel; in general, one must add to the bound from the eta invariants also the dimension of the kernel of \(D_{B_{\bullet }}\) , which can be bounded in terms of \(V,\varepsilon \)  via the local Weyl law REF . Combining this with the bound for the spectral flow REF , we conclude via Proposition .
